The church tax ("Kirchensteuer") applies if you are (declared) member of either dominant religion (Lutheran protestant or Catholic christian). It's the state which levies the tax, but only as a service to the church to which the funds are passed. It is about 10% of the income tax, i.e. about 4% of your taxable income.
I have no idea. I'd think that's something to be negotiated between church and state.
> And what if you declare yourself a member of the church but do not want to contribute - can you opt out or do you just have to lie?
If you tell the state that you're member of either of those two religions, you have to pay the tax. Tax evasion is taken seriously in Germany.
If you lie to the state, it's out of the picture. And no church will check your tax record when you go there to pray. They might refuse other services like marriage etc. if you're not declared member.
Most people though are either fond of their church and more or less gladly give or just exit. A good share of people living in Germany don't belong to any religion.
If you're single and earn well, you still might take home only about 60% due to mandatory health and social insurance contributions. Details vary, following calculator looks promising: https://www.nettolohn.de/
The real scandal is that the German state subsidizes the churches tax collection. We are not a secular state.
The historical reason is that in the 19th century the the still absolutist states in the territory of Germany took away assets from the churches that they used to fund their parishes, but they gave them the right to levy taxes as compensation.
I used to know these german taxation words, kind of, but it's a bit too late to go looking them up again.